Numerical harmonic analysis was developed before the advent of computing machines for the purpose of analyzing motions which can be adequately represented as compositions of harmonic oscillations. It was important that the analysis methods were as efficient as possible because the computations were done by hand. In this respect, Fourier analysis is the fastest method which uses the minimal amount of data to analyze such signals. Also, the first digital computers and analog to digital (A/D) converters had limited speed and resolution, so the same methods used in manual computation were developed for automated computation. In recent years the processing speeds of computers, and speed and accuracy of A/D converters have dramatically improved. However, the same harmonic analysis methods which predated the advances in computing devices and A/D converters are still in dominant use. Harmonic methods were not intended for use in conditions requiring extremely small processing delay, and are deficient when used under such conditions for intrinsic reasons.
Present day digital signal processing relies primarily on computationally intensive Fourier analysis based methods which use a relatively large number of samples taken at the Nyquist rate. Such long sequences of samples encode the "global" behavior of the input signal, i.e., as used herein, the behavior of the signal over a relatively long period of time lasting many Nyquist rate sampling intervals. However, when the signal is represented by such values, sampled at the Nyquist rate, any direct access to the information about the "local" behavior of the input signal, i.e., as used herein, the behavior of the signal over short intervals of time, in a neighborhood between two consecutive Nyquist rate samples, is lost.
Harmonic analysis represents the signal using trigonometric functions. Local variations of a signal in time are very poorly represented by periodic functions which are highly uniform in time and are suitable only for global representations of the signal over longer periods of time, i.e., many tens, or hundreds or even thousands of Nyquist rate intervals.
Not having accurate information about the local behavior of a signal makes it difficult to act in real time because, at any given instant of time, action can only be based upon the instantaneous value of the signal and values from the past. Due to the uncertainty principle, harmonic analysis requires that the signal be "seen" over a relatively long period of time, i.e. a large number of consecutive Nyquist rate samples are needed, often windowed toward the ends of the interval. Thus, harmonic analysis can only be used in instances in which delay and/or phase shifts inherent in such methods may be tolerated.
Some of the most sophisticated real time applications currently used rely on wavelet methods with cardinal splines, however, these methods still use signal processing operators which are similar to those used in harmonic analysis.
Moreover, while Nyquist's theorem enables a complete representation of a band-limited signal using only the values of the signal sampled at the Nyquist rate, if the signal is given by values sampled at the Nyquist rate only up to a time t.sub.0, then no past values between the sampling times are fully determined. If the value of the signal at a time t is approximated using the Nyquist rate values up to time t.sub.0 &gt;t, then the error of the interpolation (oversampling) depends on the energy of the signal contained in its part after t.sub.0, and no a-priori bound can be given only in terms of t.sub.0 -t, i.e. the number of samples between times t and t.sub.0. This problem is usually solved by replacing the original signal by an (overlapping) sequence of signals obtained by windowing the original signal over a sufficiently long interval of time, thus restricting the number of samples on which the interpolated values can depend.
On the other hand, Taylor's theorem provides all past and future values of the signal from the values of all derivatives of the signal at a single instant in time t.sub.0. However, Taylor's theorem is difficult to use in practice because the higher order derivatives, being extremely noise sensitive, cannot be evaluated precisely. Also, the truncated Taylor formula quickly accumulates error moving away from the point of expansion, t.sub.0. Taylor's theorem implies that the signal is determined by all of its values in the past (i.e., not just the Nyquist rate values, but values at every instant). This determinism makes such a model, even if it were feasible, inadequate for practice.
There is a need for a signal processing method and a signal processor which can characterize the local behavior of a band limited signal in terms of some suitable parameters, and which can relate such signals' local behavior parameters to the spectrum of the signal, and thus also to the global behavior of the signal. Such a method should provide a computationally effective way of achieving extremely fast responsiveness to instantaneous changes in the behavior of the signal while maintaining the spectral accuracy achieved with standard harmonic methods.